mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Cramér–Wold theorem or the Cramér–Wold device is a theorem in

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

and which states that a Borel

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...

\mathbb^k

is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold, who published the result in 1936. Let :

_n = (X_,\dots,X_)

and :

\;  = (X_1,\dots,X_k)

be random vectors of dimension ''k''. Then

_n

converges in distribution to

if and only if: :

\sum_^k t_iX_ \overset \sum_^k t_iX_i.

for each

(t_1,\dots,t_k)\in \mathbb^k

, that is, if every fixed

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of the coordinates of

_n

converges in distribution to the correspondent linear combination of coordinates of

. If

_n

takes values in

\mathbb_+^k

, then the statement is also true with

(t_1,\dots,t_k)\in \mathbb_+^k

References

Theorems in measure theory Theorems in probability theory Convergence (mathematics) {{Mathanalysis-stub